Explanation:

When a ball is thrown up in the sky and then falls back down, its motion can be divided into two phases: the ascending phase and the descending phase. In the ascending phase, the ball moves in the opposite direction to the gravitational force, while in the descending phase, it moves in the same direction as the gravitational force.

Average Velocity Calculation:
To calculate the average velocity, we need to consider the total displacement and the total time taken. Displacement is the change in position of an object and can be calculated by subtracting the initial position from the final position.

In the case of the ball being thrown up and then falling back down, the total displacement is zero because the ball returns to its initial position. The total time taken is the sum of the time taken in the ascending phase and the time taken in the descending phase.

Ascending Phase:
During the ascending phase, the ball moves against the gravitational force. As the ball moves upwards, its velocity decreases until it reaches its maximum height, where its velocity becomes zero. Since the ball is moving in the opposite direction to the gravitational force, its average velocity during this phase is negative.

Descending Phase:
During the descending phase, the ball moves in the same direction as the gravitational force. Its velocity increases as it falls towards the ground. Again, the average velocity during this phase is negative because the ball is moving downwards.

Combining Phases:
Since the ball returns to its initial position, the total displacement is zero. The total time taken is the sum of the time taken in the ascending phase and the time taken in the descending phase. Since the average velocity is calculated by dividing the total displacement by the total time taken, and the total displacement is zero, the average velocity is zero.

Therefore, the correct answer is option 'b' - the average velocity is zero.